\(H_2\)-control and the separation principle for discrete-time Markovian jump linear systems
Let \((\Omega, \{ {\mathcal F}_k\}_{k \in T}, {\mathcal F}, P)\), \(T = \{ 0, 1, \ldots\}\) be a probabilistic space. The authors consider the following Markovian Jump Linear Systems \[ {\mathcal L}_G = \begin{cases} x(k+1) = A_{\theta(k)} x(k) + B_{\theta (k)} u(k) + G_{\theta(k)} w(k), \\ y(k) = L_{\theta (k)} x(k) + H_{\theta (k)} w(k), \\ z(k) = C_{\theta (k)} x(k) + D_{\theta (k)} u(k),\end{cases} \] where \(\{x(k), k \in T\}\) represents the state vector in \(C^n\), \(\{u(k), k \in T\}\) the control sequence in \(C^m\), \(\{w(k), k \in T\}\) the input sequence in \(C^q\), \(\{y(k), k \in T\}\) the sequence of measurable variables in \(C^p\), \(\{z(k), k \in T\}\) the output sequence in \(C^r\), and the output and ``operation models'' (\(y(k)\), \(\theta (k)\) respectively) are known at each time. A \(k\)-dynamic Markovian controller \(z_k\) for \({\mathcal L}_G\) is given by \[ {\mathcal L}_k = \begin{cases} {\hat x}(k+1) = {\hat A}_{\theta(k)} {\hat x}(k) + {\hat B}_{\theta (k)} y(k), \\ u(k) = {\hat C}_{\theta (k)} {\hat x}(k).\end{cases} \] The authors show that in the case with no jumps an optimal controller can be obtained from two sets of coupled algebraic Riccati equations, one associated with the optimal control problem when the state variable is available, and the other associated with the optimal filtering problem. This is the principle of separation for discrete-time Markovian jump linear systems. The authors consider an interesting example.
scientific article; zbMATH DE number 2132326